The spectral sequence associated with a filtered chain complex is a classical notion in homological algebra, which is detailed for example in [1]}, [2]}, [3]}.
In this brief summary let us just mention that a filtration \(\lbrace  F_p C_* \rbrace _{p\in \mathbb {Z}}\)  of a chain complex has a canonically associated spectral sequence that can be explicitly constructed by
\(E^r_{p,q} = \frac{Z^{r}_{p,q} + F_{p-1}C_{k}}{d(Z^{r-1}_{p+r-1,q-r+2}) + F_{p-1}C_{k}} ,\) 
